- Write all the integers from 0 to 100 using four 4's and mathematical
symbols only. For instance, 0 = 4 + 4 - 4 - 4; 1 = 44/44; and so on.
(Symbols: +, -, *, /, (, ), !, sqrt, ^, .)
- 48 is an interesting number. If you add 1, you get 49, a perfect
square. If you divide it by 2 and then add 1, you get 25, another
perfect square. What is the next integer with this property?
- You are given 13 coins, 12 genuine and 1 counterfeit. It is given that the
counterfeit coins and genuine coins have different weights. You are also
given a scale which when given a pair (A,B) of disjoint subsets of the 13
coins returns how the weights of A and B compare (equal, greater, or less).
If you are allowed at most 3 uses of the scale, how would you determine which
coin is counterfeit? Can you generalize this problem? (With an appropriately
generalized solution, of course!)
- Find all the total functions f that map reals to reals that are not identically zero, and that satisfy the following relation: f((x+y)/(x-y)) = (f(x)+f(y))/(f(x)-f(y))
- A chessboard can be tiled with 32 dominos. Suppose we remove the left
top and right bottom corner of the board. Can the remaining 62 squares be
tiled by 31 dominos?
- You are given a bar of chocolate with 50 squares (5 x 10). What is
the minimum number of breaks necessary to break the bar into 50 individual
squares? A bar (or a piece of it) can only be broken along a straight line
that runs from one side to the other. Pieces cannot be stacked while breaking.
- Given two positive numbers, what is the probability that they are
relatively prime? (Assume a uniform distribution.)
- Let f be a total function from natural numbers to natural numbers.
It has the property that f(f(n)) < f(n+1) for all n. Find a simple
characterization of f.
- Show that a path on an m by n square grid which starts at the northwest
corner, goes through each point exactly once, and ends at the southeast
corner divides the grid into two equal halves: (a) those regions opening
north and east; and (b) those regions opening south or west.
- Adam, Rajit, Rob and Rohit are sitting on the beach at Malibu around 3am.
Rajit, what were the numbers?
- "I just picked two integers greater than one."
- "Adam, their sum is ..." (he whispers it to Adam).
- "Rohit, their product is..." (he whispers it to Rohit).
- "Rohit, we don't know the numbers."
- "Now I do."
- "Me too".
- Person A and B play the following game. Person A makes a bet of x dollars
on heads or tails. Person B then tosses a fair coin. If the outcome matches A's
guess, A receives 2x dollars; otherwise, A loses his bet. Consider the
following strategy for A: x:=1 initially; if win, then x:=1; otherwise
x:=x*3. Now, if A wins at any point, s/he makes a profit. This means that
A will always make a profit. What (if anything) is wrong with this strategy?
- There are 100 light bulbs and 100 people, both numbered from 1 to 100.
Initially, all the light bulbs are off. Person number k toggles all the
light bulbs that are divisible by k. For example, person 2 toggles bulbs
2, 4, 6, ..., 100. After all 100 people have finished toggling the light
bulbs, which light bulbs are on?